CLAP - I. Resolving miscalibration for deep learning-based galaxy photometric redshift estimation

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Volume 691 (November 2024)

A&A, 691 (2024) A331

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Table 2

Estimates and summary statistics derived from probability densities.

Point estimate z_photo	$\int_{0}^{z_{m a x}} z \times p (z ∣ d) d z$ $\int_0^{{z_{max}}} z \times p(z\mid d){\rm{d}}z$
Cumulative density estimate F(z\|d)	$\int_{0}^{z} p (z^{'} ∣ d) d z^{'}$ $\int_0^z p \left( {z'\mid d} \right){\rm{d}}z'$
Instance-wise residual δ-	(z_photo − z_spec)\|(1 + Z_spec)
Bin-wise mean redshift residual δ_<z>	(< Z_photo > − < Z_spec >)/(1+ < Z_spec >) ^(*)
MAD-based deviation σ_MAD ^(**)	1.4826 × Median \|δ_z – Median(δ_z)\|
Probability integral transform (PIT)	$\int_{0}^{z_{spec}} p (z ∣ d) d z$ $\int_0^{{z_{{\rm{spec }}}}} p (z\mid d){\rm{d}}z$
1-Wasserstein distance with the one-hot label	$\int_{0}^{z_{spec}} F (z ∣ d) d z + \int_{z_{spec}}^{z_{\max}} \| 1 - F (z ∣ d) \| d z$ $\int_0^{{z_{{\rm{spec }}}}} F (z\mid d){\rm{d}}z + \int_{{z_{{\rm{spec }}}}}^{{z_{\max }}} \| 1 - F(z\mid d)\|{\rm{d}}z$
Continuous ranked probability score (CRPS)	$\int_{0}^{z_{s p e c}} F {(z ∣ d)}^{2} d z + \int_{z_{s p e c}}^{z_{\max}} (1 - F (z ∣ d))^{2} d z$ $\int_0^{{z_{{\rm{spec }}}}} F {(z\mid d)^2}{\rm{d}}z + \int_{{z_{{\rm{spec }}}}}^{{z_{\max }}} {(1 - F(} z\mid d){)^2}{\rm{d}}z$
Cross-entropy with the one-hot label	− log p(z_spec\|d)
Entropy	$- \int_{0}^{z_{\max}} p (z ∣ d) \log p (z ∣ d) d z$ $- \int_0^{{z_{\max }}} p (z\mid d)\log p(z\mid d){\rm{d}}z$
Standard deviation σ	$\sqrt{\int_{0}^{z_{\max}} {(z - z_{p h o t o})}^{2} p (z ∣ d) d z}$ $\sqrt {\int_0^{{z_{\max }}} {{{\left( {z - {z_{photo}}} \right)}^2}} p(z\mid d){\rm{d}}z}$
Skewness	$\int_{0}^{z_{\max}} {(z - z_{photo})}^{3} p (z ∣ d) d z / σ^{3}$ $\int_0^{{z_{\max }}} {{{\left( {z - {z_{{\rm{photo }}}}} \right)}^3}} p(z\mid d){\rm{d}}z/{\sigma ^3}$
Kurtosis	$\int_{0}^{z_{\max}} {(z - z_{photo})}^{4} p (z ∣ d) d z / σ^{4}$ $\int_0^{{z_{\max }}} {{{\left( {z - {z_{{\rm{photo }}}}} \right)}^4}} p(z\mid d){\rm{d}}z/{\sigma ^4}$

Notes.^(*) <> denotes the mean computed in a redshift or magnitude bin.^(**) MAD stands for the median absolute deviation.

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